application of fastnlo to nnlo calculationsrabbertz/office/physics_talks/pd… · ->...

Application of fastNLO to NNLO calculations

DIS 2014 XXII. International Workshop on Deep-Inelastic Scattering and Related Subjects

April 30, 2014

Daniel Britzger , Marco Guzzi, Klaus Rabbertz, Georg Sieber, Fred Stober,Markus Wobisch(DESY, DESY, KIT , KIT, KIT, Louisiana Tech)

2

Motivation

Interpretation of experimental data relies on• Availability of reasonably fast theory calculations• Often needed: Repeated computation of (almost) same cross sections

• Observables, binning, phase space given by experimental data

Examples for a specific analysis:• Use of various PDFs (CT, MSTW, NNPDF, …) for data/theory comparison• Determine PDF uncertainties• Derivation of scale uncertainties• Use data set in fit of PDFs and/or αs(MZ)

Sometimes NLO predictions can be computed fastBut some are very slow

e.g. jet cross sections, VB+jets, Drell-Yan, …

Need procedure for fast repeated computations of NL O cross sections• Use fastNLO (in use by most PDF fitting groups)

3

Basic working principle Cross section in hadron-hadron collisions in pQCD

• Many cross section calculations are time consuming (e.g. jets)

• strong coupling αs in order n• PDFs of two hadrons f1, f2

• Parton flavors a, b

• perturbative coefficent ca,b,n

• renormalization and factorization scales µr, µf

• momentum fractions x1, x2

),(),(),,,()( 2,21,121,,,,

1

0

2

1

0

1 fbfafrnbanba

rns xfxfxxcdxdx µµµµµασ ⋅⋅= ∑ ∫∫

PDF and αs are external inputPerturbative coefficients are independent from PDF and αs

Idea: factorize PDF, αs and scale dependence

c

f1(x1)

f2(x2)

4

The fastNLO concept Introduce interpolation kernel

• Set of n discrete x-nodes xi's being equidistant in a function f(x)

• Take set of Eigenfunctions Ei(x) around nodes xi-> interpolation kernels

Single PDF is replaced by a linear combination of interpolation kernels Improve interpolation by reweighting PDF

Scale dependence• Similar interpolation procedure also for scales• Interpolation nodes in x and scales are stored

together in look-up table• Two different observables can be chosen as

scale in one table

)()()( )( xExfxf ii

iaa ⋅≅ ∑

Convolution integrals become discrete sums => Values of perturbative coefficents can be stored in a table

5

Calculations with fastNLO in NNLO Problem

• Scale variations become more difficult in NNLO than in NLO

Current available implementations for NLO calculations Renormalization scale variations

• Scale variations applying RGE• Use LO matrix elements times nβ0ln(cr) [fastNLO, APPLgrid (EPJ C (2010) 66: 503)]

• Flexible-scale implementation• Store scale-independent weights: [fastNLO]

Factorization scale variations• Calculate LO DGLAP splitting functions using HOPPET [APPLgrid]• Store coefficients for desired scale factors [fastNLO]• Flexible-scale implementation [fastNLO]

Scale variations for NNLO calculations• a-posteriori renormalization scale variations become more complicated• NLO splitting functions are needed for factorization scale variations

• Calculations become slow again => Not desired for fast repeated calculations

FFRRFR ωµωµωµµω )log()log(),( 0 ++=

6

Flexible-scale implementation in NNLO Storage of scale-independent weights enable full sc ale flexibility also in NNLO

• Additional logs in NNLO

• Store weights: w0, wR, wF, wRR, wFF, wRF for order αsn+2 contributions

Advantages• Renormalization and factorization scale can be varied independently and by any factor

• No time-consuming ‘re-calculation’ of splitting functions in NLO necessary

• Only small increase in amount of stored coefficients

fastNLO implementation• Two different observables can be used for the scales

• e.g.: HT and pT,max

• or e.g.: pT and |y|

• …

• Any function of those two observables can be used for calculating scales

‘Flexible-scale concept‘: Best choice for performant NNLO calculations

RFFFRRFRFR FRFRFRωµµωµωµωµωµωµµω )log()log()(log)(log)log()log(),( 22222222

0 +++++=

additional log’s in NNLOlog’s for NLO

7

0.5

1

1.5approx. NNLO, m t=173 ±1 GeV

d

σ/dp

T (

pp→

tt– +X)

(pb/

GeV

)

CT10NNLOMSTW08NNLOABM11NNLO

HERAPDF1.5NNLO

NNPDF2.3NNLO

pt T (GeV)

σ/σ C

T10

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350 400

Application to differential ttbar cross sections in approx. NNLO

Application of flexible-scale concept to NNLO calculations

Interface to DiffTop code • DiffTop

• Code for calculation of heavy quark production within threshold resummation formalism in Mellin space

• See talk by M.Guzzi

• Differential ttbar cross sections in approx. NNLO• dσ/dpT

• dσ/dy

Benefit in speed• NNLO calculation O(days-weeks)• fastNLO calculation O(<1s)

fastNLO facilitates study of PDF dependence Particularly including PDF uncertainties

• 262 re-calculations are required

preliminarysee talk by M. Guzzi

786 repeated calculations needed including variation of mt

8

Application to differential ttbar cross sections in approx. NNLO

Without recalculating the coefficients• Variation of the scales within milliseconds

• Variation of αs

• Determination of PDF uncertainties• Also choice of the scales possible

here: µr/f = f(pT,y,mt)

Variation of mt

• mt is a third hard scale in this process• mt is not factorized in the current approach

• Separate fastNLO tables have been computed for different values of mt

Fast recomputation of cross sections for a given measurement enables application of time-consuming (N)NLO calculations to PDF and/or αs-fits

• Large gluon uncertainties at high-x can be reduced using ttbar cross sections

20

40

60

80

100approx. NNLO × CT10NNLO, m t=173 GeV

d

σ/dy

(pp

→tt– +X

) (1

/GeV

)

total uncertainty

δmt= 1 GeV

PDF 68%CL

µr=µf var.

αS

yt

rela

tive

erro

r

0.8

0.9

1

1.1

1.2

-3 -2 -1 0 1 2 3

dσ/dy/σ (pp→tt–+X) , mt=173 GeV

δmt= 1 GeVPDF 68%CLµr=µf var.αS

approx. NNLO × CT10, total unc.

data CMS, √s=7 TeV

yt

theo

ry/d

ata,

dσ/

dy/σ

0.8

0.9

1

1.1

1.2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

20

40

60

80

100approx. NNLO × CT10NNLO, m t=173 GeV

d

σ/dy

(pp

→tt– +X

) (1

/GeV

)

total uncertainty

δmt= 1 GeV

PDF 68%CL

µr=µf var.

αS

yt

rela

tive

erro

r

0.8

0.9

1

1.1

1.2

-3 -2 -1 0 1 2 3

dσ/dy/σ (pp→tt–+X) , mt=173 GeV

δmt= 1 GeVPDF 68%CLµr=µf var.αS

approx. NNLO × CT10, total unc.

data CMS, √s=7 TeV

yt

theo

ry/d

ata,

dσ/

dy/σ

0.8

0.9

1

1.1

1.2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

9

Accuracy of fastNLO interpolation in NNLO

Compare cross sections calcuated with DiffTop standalone compared to fastNLO

• Interpolation accuracy depends on number of nodes and on chosen interpolation kernel

• Bicubic interpolation kernels used• Compare contribution order by order separately• Data probe x-range of 2 ·10-3 < x < 1

• High x-range has more distinct PDF shapes

fastNLO/DiffTop• Perfect agreement within numerical precision

reachable (O(10-6))• NNLO has same accuracy as LO• With 18 nodes agreement better than 2·10-4

• Accidental ‘interference effects’ with PDF grid may cause small fluctuations O(2·10-4) -> Numerical uncertainty of PDF grids

fastNLO interpolation does not introduce numerical biases

Number of x-nodes0 10 20 30 40 50 60 70 80

fast

NLO

/ D

iffT

op

0.9995

1

1.0005

1.001

Interpolation precision LO

= 10 GeVtT

p

= 80 GeVtT

p

= 120 GeVtT

p

= 600 GeVtT

p

Number of x-nodes0 10 20 30 40 50 60 70 80

fast

NLO

/ D

iffT

op0.9995

1

1.0005

1.001

Interpolation precision NNLO

= 10 GeVtT

p

= 80 GeVtT

p

= 120 GeVtT

p

= 600 GeVtT

p

Interpolation precision NNLO

MSTW2008nnlo

MSTW2008nnlo

10

New tool: fastNLO toolkit

What about application of fastNLO to other processe s/programs ? Hardly any theoretical limitation of fastNLO concept to pQCD or EW calculations

Why not used more frequently?

Interface of fastNLO to theory programs often very complicated…• Theory codes are not optimized (at all) for fastNLO • Technical difficulties are mostly limiting factor in usage

Goal:Provide simple and flexible code to interface fastN LO to any kind of (N)NLO program

Newly developed tool: fastNLO Toolkit

11

Application procedure of new ‘fastNLO Toolkit’

fastNLO needs access to• Matrix elements before convolution with PDFs• x-values where PDFs are evaluated• Observables• Scale definitions

Various NLO and NNLO programs have very different software architecture

Reasons• Optimized for efficient calculation• Straight implementation of math. formulae• Historically grown codes• Usage of well established algorithms (e.g. vegas)

(N)NLO programs often look to non-authors like different kind of pasta

12






NLO program A

CLO CNLO

µrPDF

Obs1

13






NLO program B

NLO program A

CLO CNLO

µrPDF

Obs1

CLO

CNLO

µr

PDF

Obs1

14






NLO program B

NLO program A

CLO CNLO

µrPDF

Obs1

CLO

CNLO

µr

PDF

Obs1

Think about a general interface to any kind of theoretical program

15


(N)NLO Program

MC Integration

Program Start

Program End

(N)NLO Result

16


Initialize fastNLO class(es)(N)NLO Program

fastNLOCreate fnlo(„steering.str“);fnlo.SetOrderOfCalculation(int order);

MC Integration

Program Start

Program End

(N)NLO Result

17




fnlo.fEvent.SetProcessID(int id);

fnlo.fEvent.SetWeight(double w);

fnlo.fEvent.SetX1(double x1);fnlo.fEvent.SetX2(double x2);

fnlo.fScenario.SetObservable0(double pt);fnlo.fScenario.SetObsScale1(double s1);

fnlo.Fill();

MC Integration

Program Start

Program End

(N)NLO Result

Pass the process specific variables during the ‘event loop’to fastNLO

• Order does not matter• Many other convenient

implementations possible

Pass all information to fastNLO

18




fnlo.fEvent.SetProcessID(int id);

fnlo.fEvent.SetWeight(double w);

fnlo.fEvent.SetX1(double x1);fnlo.fEvent.SetX2(double x2);

fnlo.fScenario.SetObservable0(double pt);fnlo.fScenario.SetObsScale1(double s1);

fnlo.Fill();

fnlo.SetNumberOfEvents(double nevents);fnlo.WriteTable();

MC Integration

Program Start

Program End

(N)NLO Result fastNLO Table

Set normalization of the MC integration and write table

Pass the process specific variables during the ‘event loop’to fastNLO

• Order does not matter• Many other convenient

implementations possible

Convenient implementation of fastNLO into any (N)NLO program possible

Pass all information to fastNLO

Minimum implementation:11 lines of code

19

New developments for fastNLO toolkit

1) Creation of fastNLO tables• One stand-alone c++ library

• No third party packages needed• Optimally: Only 11 lines of code necessary • Depending on (N)NLO program:

many implementations possible

• Parameters are specified in steering card• Binning (also double- or triple- differentially)

• Performance optimized• Caching of interpolation values • Automatic optimization of grids to phase

space

• PDF parton combinations for different subprocesses• Specified in steering• Stored in table

2) fastNLO table format• Further contributions are forseen

• EW corrections, etc…

• PDF combinations are stored in table• Storage of uncertainties soon available• QEDPDFs or p-Pb collisions also forseen

3) Evaluating fastNLO tables• New interface in PYTHON

• Many interfaces to PDF and αs routines• LHAPDF5, LHAPDF6, Hoppet, QCDNUM,

ALPHAS, CRunDec, …

• Improved speed for flexible-scale tables• Caching of PDFs and αs values for (even)

faster re-evaluation

// FastNLO example code in c++ for reading CMS incl.

// jets (PRL 107 (2011) 132001) with CT10 PDF set

fastNLOLHAPDF fnlo("fnl1014.tab","CT10.LHgrid",0);

fnlo.PrintCrossSections();

20

Summary and Outlook

fastNLO enables fast re-evalution of perturbative c alculations• Convenient for scale or PDF studies (e.g. uncertainties)• Mandatory tool for phenomenological analysis (e.g. αs or PDF fits)

Scale-independent concept successfully applied in N NLO

fastNLO is applicable to a wide range of processes and corrections

New tool: ‘fastNLO toolkit’• Facilitates creation of tables and interface to other (N)NLO programs• Very flexible code:

Only few modifications in (N)NLO programs are needed (O(11) lines of code)

Code and manual is released soon after conference• Python interface available for reading tables

• pre-release version of ‘fastNLO toolkit’ is available on request• more information: http://fastnlo.hepforge.org

22

Application of flexible scale concept Inclusive jet production in DIS

Two scales are stored in table• Q2

• pT of the jet Any function of the two can be used as scale

Renormalization and factorization scale can be varied seperately

Choose for scale study• µr

2 = Q2+pT

• µf2 = Q2

• Color code shows 5% change in cross section w.r.t. to scale factor of 1

Other scale choices also shown without scale factors

QCD compton QCD compton Boson− gluon fusion Boson− gluon fusion

application of fastnlo to nnlo calculationsrabbertz/office/physics_talks/pd… · ->...

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